Mayim SheLanu = Our Water?

There is some wonderful alliteration in Bereishit 24:23

וַיֹּאמֶר בַּת-מִי אַתְּ, הַגִּידִי נָא לִי; הֲיֵשׁ בֵּית-אָבִיךְ מָקוֹם לָנוּ, לָלִין.
"and said: 'Whose daughter art thou? tell me, I pray thee. Is there room in thy father's house for us to lodge in?'"

in the well-crafted phrase לָנוּ לָלִין.

לָנוּ mean "for us," for we see later that Eliezer had accompaniment.
Bereishit 24:54:

וַיֹּאכְלוּ וַיִּשְׁתּוּ, הוּא וְהָאֲנָשִׁים אֲשֶׁר-עִמּוֹ וַיָּלִינוּ; וַיָּקוּמוּ בַבֹּקֶר וַיֹּאמֶר שַׁלְּחֻנִי לַאדֹנִי.
"And they did eat and drink, he and the men that were with him, and tarried all night; and they rose up in the morning, and he said: 'Send me away unto my master.' "

where we have וַיָּלִינוּ, and not as I idly speculated, that לָנוּ might be from the same root as לָלִין.