Psalm 116 - Laudate Dominum

Psalm 116 is the shortest in the psalter, so short that St Benedict doesn't even treat it as a separate psalm in his ordering of the psalter, but adds it the end of Psalm 115 under the same Gloria.

Laudáte Dóminum, omnes Gentes: * laudáte eum, omnes pópuli :
Quóniam confirmáta est super nos misericórdia ejus: * et véritas Dómini manet in ætérnum.

or 

Praise the Lord, all you nations: praise him, all you people. 
For his mercy is confirmed upon us: and the truth of the Lord remains for ever.

Laudáte (praise) dóminum (the Lord), omnes (all) gentes (peoples): * laudáte (praise) eum (Him), omnes (all) pópuli (peoples)

Quóniam (for) confirmáta est (it is confirmed/established) super (upon/over) nos (us) misericórdia (mercy) ejus (his): * et (and) véritas (the truth) dómini (of the Lord) manet (it remains/endures) in ætérnum (forever)

Why is this psalm so suitable for Monday Vespers, and why is it joined to Psalm 115?  

Firstly, like Psalm 115, this psalm links us back to Psalm 2, as St Robert Bellarmine comments:

He addresses the whole Church and exhorts it to praise God.  “All ye nations” is directed to the converted Gentiles, who are named first by reason of their being in the majority, and the people nearer those of the Jews who had been converted to the faith; and the apostles themselves, in alluding to a similar expression in the second Psalms, “Why have the Gentiles raged, and the people meditated vain things," apply the former to the Gentiles, and the latter to the Jews.

Secondly, it brings us back, once more, to our response to the salvation offered through the Incarnation and Christ's public ministry, as Cassiodorus points out:

The reason is given why the Lord must be praised throughout the world: it is because He has fulfilled His promises made through the holy prophets by His coming to us. His mercy towards the Christian people is confirmed and will not be moved for ever, for He who granted it, as we justly believe, protects us with His pity. He added: And the truth of the Lord remaineth for ever. The truth of the Lord here means the Son; as He Himself says: I am the way, the truth and the life? 

In the notes on verse 7 of Psalm 115, I quoted Pope Benedict on the importance of the 'sacrifice of praise' offered by religious for the world as testimony.  In the context of this Psalm, though, it is worth considering Pope Benedict's other comments on the reasons for this offering:

Monks pray first and foremost not for any specific intention, but simply because God is worthy of being praised. “Confitemini Domino, quoniam bonus! – Praise the Lord, for he is good, for his mercy is eternal!”: so we are urged by a number of Psalms (e.g. Ps 106:1). Such prayer for its own sake, intended as pure divine service, is rightly called officium. It is “service” par excellence, the “sacred service” of monks. It is offered to the triune God who, above all else, is worthy “to receive glory, honour and power” (Rev 4:11), because he wondrously created the world and even more wondrously renewed it. (Speech at Heiligenkreuz Abbey, 2007)

Scriptural and liturgical uses of the psalm

NT references	Rom 15:11 (v1);
RB cursus	Mon V
Monastic feasts etc	-        (NB in Roman feasts, Vespers)
Roman pre 1911	Monday Vespers
Responsories	7078 (v1, Romanum; cf Sept, for Judith)
Roman post 1911	1911-62: Monday Vespers . 1970:
Mass propers (EF)	Lent, Pent & Sept Ember Sat TR 1-2; Easter Vigil AL (1-2),

And you can find notes on the final psalm of Monday Vespers Psalm 128, here.