How to only update 1 row instead of all the rows when clicking a button

To help your injection problem, parameterize. It would be something like this (I use PDO, so you will want to double check):

/functions/getUtleie.php

function getUtleie($so, $conn)
{
    $query = $conn->prepare("SELECT utleid, inntid, baatnr, fornavn, etternavn, tid, kr, baatinn FROM utleie WHERE baatnr LIKE ? or fornavn LIKE ? or etternavn LIKE ? or tid LIKE ? ORDER BY id desc");
    $so = "%{$so}%";
    $query->bind_param('ssss',$so, $so, $so, $so);
    $result = $query->execute();
    if($result->num_rows == 0)
        return [];

    while($row = $result->fetch_assoc()) {
        $data[] = $row;
    }

    return $data;
}

Now, when you go to use it, include the function, then the key on the form is to make the id in a hidden field:

# Fetch the data
$result = getUtleie($so, $conn);
# If there are any results
if(!empty($result)): ?>
    <table>
    <?php foreach($result as $row): ?>

        <tr>
            <td><?php echo $row["utleid"] ?></td>
            <td><?php echo $row["inntid"] ?></td>
            <td><?php echo $row["baatnr"] ?></td>
            <td><?php echo $row["fornavn"] ?></td>
            <td><?php echo $row["etternavn"] ?></td>
            <td><?php echo $row["tid"]; ?></td>
            <td><?php echo $row["kr"]; ?></td>
            <td>
                <form method="post" action="innlevering.php">
                    <input type="hidden" name="action" value="update_utleie" />
                    <input type="hidden" name="utleid" value="<?php echo $row["utleid"] ?>" />
                    <input type="text" name="val" />
                    <input type="submit" value="Edit" />
                </form>
            </td>
        </tr>

        <?php endforeach ?>

    </table>

<?php else: ?>

0 results

<?php endif ?>

After you submit the form, you will want to update using a WHERE clause:

<?php
include_once 'dbconnect.php';
# Check to make sure the form was submitted
if(!empty($_POST['action'] && $_POST['action'] == 'update_utleie') {
    # Trim these. You should also check they aren't empty (especially the id)
    $id    = trim($_POST['utleid']);
    $value = trim($_POST['val']);
    $query = $conn->prepare("UPDATE `utleie` SET `baatinn` = ? WHERE `utleid` = ?");
    $query->bind_param('si', $value, $id);
    $query->execute();
}

Anyway, I haven't checked these scripts but it should be pretty close. Should at least point you in the right direction.