Besides this askbot site, there is also a forum on open office suites (including LibO) where you can find solutions.
In your case: [Tutorial] Randomization in Calc (View topic) • Apache OpenOffice Community Forum
“C.” “E.” (Sorry!) and the example document attached there apply to your question.
Though the OQ didn’t answer my questions I prepared this attached demo.
You may take from it what seems useful to you.
(Edit2 regarding the recent explanations by the OQ:)
To create a new partition-into-pairs is simple using the means I already demonstrated, but I would judge it next to impossible (probable solutions at least very inefficient) to definitely avoid any reoccurring pair by Calc means. If a respective guarantee is needed I would resort to user code. (Also not exactly simple. These comments are based on estimations, of course.)
(deleted; was not good)
Now hopefully better:
What you actually need has little do do with randomizing. Depending on the preferred words it is nown as a chapter of mathematical graph-theory, or more pragmatically - often needed - the task to schedule (sine-die) the first set of matchdays for a 20-league. In your case the participants are just not expected to fight one another.
Spreadsheets for the task: http://www.excelformeln.de/bilder/Liga-Spielplan.zip
(German; no understanding of the language needed. Simply replace the names.)
(Author’s page in that site: http://www.excelformeln.de/uberuns.html?umenuzahl=14&autor=1)
If randomization shall be included you may first of all randomize the order of names (and leave it at that then) or randomize the order of the pairings you want to apply them in (or both).
The linked sheet is in .xls and contains relics of VBA though there is no active code at all. I made an .ods of it without the macro modules (“hard work”) and it still did its duty. Of cours you may simply reject permission for macros.